Multiple vacua for non-abelian lattice gauge theories

Abstract The various formulations of gauge theories characterized by the parameter θ are constructed for the lattice version of these theories. We do not rely on the existence of topologically stable solutions of the classical equations. These constructions are based on the existence of inequivalent representations of the canonical commutation relations.


Introduction
The Hilbert space spanned by the eiegnstates of a Hamiltonian invariant under a discrete set of transformations, T n, may be decomposed into a sum of subspaces, such that in each subspace the states transform with a definite phase, Tla;O) = e 2~/° la;0). (1.1) As long as all allowed operators (observables) commute with T, the different 0 sectors do not communicate with each other. On the classical level such a periodicity is associated with topologically stable solutions of the corresponding Euclidean theory. The discovery of such solutions [ 1 ] for Yang-Mills theories has focussed attention on the existence of such 0 sectors in these models [2]. A different approach to the study of these theories has been to work in a spacetime or space lattice [3,4], rather than in a continuum. We address ourselves to the question how to generalize the expressions for topological quantities to the lattice and thus introduce the different 0 theories. It is important to maintain invariance under local gauge transformations.
Our approach will be based on a sequence of observations. In continuum situations the quantum mechanics of a particular 0 sector of a theory specified by an * Work supported in part by US Department of Energy under Contract EY76-C-03-0068. ** Permanent address. 211 action So is equivalent to the 0 = 0 sector of another theory specified by art action So -OS1. S~ is related to topological invariants; a property which we shall not use here. The property of interest is that S~ is a time integral over a total time derivative. On a classical level S o and So -OS] are related by canonical transformations. On a quantum level, such a transformation is achieved by different realizations of canonical commutation relations. It is this property we shall use in obtaining different 0 theories on a lattice. This observation, that reference to Euclidean solutions or topological invariants is not necessary for the understanding of multiple vacua, has been made previously in the case of two-dimensional Abelian theories [5].
The difficulty in extending these ideas to lattice non-Abelian theories is that it is not immediately possible to write the generator of the aforementioned contact transformations, or the different realizations of the quantum mechanical commutation relations, in a form invariant under local gauge transformations. The generator of the contact transformations in the continuum case must first be cast into a form manifestly invariant under local gauge transformations, with terms affected only by global transformations appearing explicitly as surface integrals. A lattice analogue can then be obtained immediately.
The fact that differing physical systems may have seemingly identical Hamiltonians, as an operator is defined only when its domain is likewise specified, is illustrated by the simple example of a one-dimensional periodic potential and continuum gauge theories. No new results are obtained as this serves as an introduction to our methods. A crucial identity permitting the extension of continuum methods to the lattice is discussed in subsect. 3.2. ff(~b + 2rr) = e 2~ri° ff(~b).

One-dimensional periodic potential
(2. 3) The spectrum ofp now consists of n + 0, where n is an integer. This new realization can be viewed in a somewhat different manner. The only a priori relation we know between p and ~b is the commutation relation Eq, (2.2) is not the only realization of such commutation relations, for we may make the identification p =-i~+ 0, (2.5) and operate on periodic functions. (0 could be a function of q~, but we will not be concerned with this more general situation.) Another way of obtaining eq. (2.5) is first to perform a contact transformation on the classical Hamiltonian, p~P=p+O , q~ ~ q5 = q~, (2.6) and then replace p by the realization of eq. (2.2). The generating function for the above transformation is [6] F2(P, (P) = c~P -00.
The action corresponding to the Hamiltonian acquires an additional term: Our reason for belaboring these rather trivial points is that finding the analogues of eq. (2.7) is a sure way of finding different realizations of the basic commutation relations without upsetting any of them. Bypassing the classical level, this new realization may be obtained by the transformation Despite appearances, U(40 is not an unitary transformation for it connects different Hilbert spaces.
In order to fix the theory completely, we must specify 0 in addition to specifying the Hamiltonian. To determine some relations between the differing 0 worlds it is useful to study the evolution kernel for the various theories corresponding to a given Hamiltonian. Let us first consider the situation where the position variable ¢ is unconstrained and allowed to vary between plus and minus infinity. For this case let It is easy to check that for the periodic case where -~r < q5 < lr and a given 0 we obtain Go(dp', dp; t) = ~e 2"in° G(¢' + 21rn, ¢; t) . (2.11) It was noted in sect. 1 that as long as we are interested in operators having the periodicity of the potential, the different 0 worlds do not communicate with each other. If on the other hand, our algebra of operators includes non-periodic ones, the Hilbert space of states consists of a sum of the 0 spaces. Eq. (2.11) may be inverted, Finally, note that the operator U(q~) of eq. (2.9) is not periodic; part of the difficulty in the construction of lattice gauge theories will be in the determination of such aperiodic variables.

Continuum theory
As the easiest formulation of a lattice gauge Hamiltonian is in the Ao = 0 gauge, we will, likewise, discuss the continuum theory in this gauge. With Integration by parts yields = -g/'dr Tr mo. E + g Tr/'dS" EA.

Reformulation of W[A ]
All the previous discussion is standard and served as an introduction to the procedure for lattice theories. To follow the previous steps we will need the lattice analogue of W [A]. Besides requiring that it has the correct continuum limit, we also insist that it be invariant under local lattice gauge transformations. The form of IliA] given in eq. (3.8) does not have an obvious lattice version satisfying the above criteria.
In this section we shall obtain an expression for IV[A] consisting of a volume term which is manifestly invariant under all gauge transformations and a surface term which changes appropriately under global gauge transformations.
Let us separate the coordinates of each point into its z-component and a twodimensional vector ri, perpendicular to z. Define v-l(r±,z)= P exp j A3(h,z' ) dz , (3.14) __oo where P denotes a path ordering. It is now a matter of integration by parts to show that 87r2W[ A ] = !e"2 ~1 Tr fd3ri dz dz'e(z' -z) The first term, a volume integral is manifestly gauge invariant. The next two terms are surface integrals while the last, although at first glance appears to depend on all of space, can be brought to the form of a surface integral. This is obvious in that this term has the form of the winding number of a pure gauge field and therefore depends only on the value of its variables on the surface. More specifically, note that V is a unitary matrix and therefore may be written as ei] Tr f dr(g-~iv)(g-l~jg)(g-l~3g) = fdSkeki](l[3 -¼ sin 2t3)~" (8i~ ^ 0/~). (3.17) As in the one-dimensional case we have to consider functions of "angles" rather than of periodic variables.

Lattice theory
Hamiltonian lattice gauge theories may be viewed as a collection of interacting symmetric tops defined on the links of the lattice [4]. At each lattice point r we associate a set of right-handed coordinate axes and denote the respective links by (r, i). On each link we define the matrix u(r, i) = exp aAi(r),   Lagrangian. This would have the correct continuum form, but for any finite lattice constant this term would not be a time derivative, or even a lattice finite time difference.
We have no answer to the question whether the continuum limits of the two quantum theories are the same. As a last point we mention that, as in eq. (2.12), we may integrate over 0 in the evolution kernel and obtain a theory with no periodicity at all.